Orthogonality and Orthonormal Sets Calculator

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Example

Created on 2024-06-20Asked by Henry Carter (Solvelet student)

Determine if the set of vectors

\mathbf{u}_1 = (1, 0, 0)

\mathbf{u}_2 = (0, 1, 0)

, and

\mathbf{u}_3 = (0, 0, 1)

is orthonormal.

Solution

To determine if the set of vectors

\mathbf{u}_1 = (1, 0, 0)

\mathbf{u}_2 = (0, 1, 0)

, and

\mathbf{u}_3 = (0, 0, 1)

is orthonormal: 1. Check if the vectors are orthogonal by computing the dot products:

\mathbf{u}_1 \cdot \mathbf{u}_2 = (1)(0) + (0)(1) + (0)(0) = 0,

\mathbf{u}_1 \cdot \mathbf{u}_3 = (1)(0) + (0)(0) + (0)(1) = 0,

\mathbf{u}_2 \cdot \mathbf{u}_3 = (0)(0) + (1)(0) + (0)(1) = 0.

2. Check if each vector is a unit vector:

\| \mathbf{u}_1 \| = \sqrt{(1)^2 + (0)^2 + (0)^2} = 1,

\| \mathbf{u}_2 \| = \sqrt{(0)^2 + (1)^2 + (0)^2} = 1,

\| \mathbf{u}_3 \| = \sqrt{(0)^2 + (0)^2 + (1)^2} = 1.

Since the vectors are mutually orthogonal and each has a magnitude of 1, the set of vectors is orthonormal. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Eleanor Martin on Solvelet

1. Construct an orthonormal basis for the subspace spanned by the vectors

[1, 0, 1]

and

[1, 1, 0]

.2. Use the Gram-Schmidt process to orthogonalize the vectors

[1, 1, 0]

and

[0, 1, 1]

\mathbb{R}^3

DefinitionOrthogonal sets in which each vector has unit length are called orthonormal sets. These ideas are important in linear algebra and functional analysis. Example: A set {(1,0),(0,1)} are orthonormalization because, they are orthogonal, and each has length of 1.

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